This is a 2-part, proof-of-concept study to be conducted globally, designed to evaluate the safety, tolerability, clinical activity, pharmacokinetics, and pharmacodynamics of BLU-808, a wild type KIT inhibitor, in participants with CIndU (Part A) or CSU (Part B).

Key Inclusion Criteria:

• Part A: Confirmed diagnosis of CIndU for ≥3 months prior to Day 1 that is inadequately controlled with second generation H1-antihistamines.
• Part B: Confirmed diagnosis of CSU for ≥3 months prior to Day 1 that is inadequately controlled with second generation H1-antihistamines.

Key Exclusion Criteria:

• Part A: Any active urticaria that may interfere with study assessments.
• Part B: Participant has a clearly defined predominant cause of chronic urticaria or sole trigger such as symptomatic dermographism and cold-induced urticaria.
• Part A and Part B: Any other skin disease associated with chronic itching or angioedema that might influence the study evaluations and results, skin diseases associated with only wheals and no itch, or autoinflammatory diseases with urticarial lesions.
• Part A and Part B: Significant medical, psychiatric, or surgical conditions, or physical findings that may affect participant safety, study drug metabolism, study participation, or assessment of study results.
• Part A and Part B: Abnormal laboratory values that may pose risks or interfere with study participation.
• Part A and Part B: Pregnancy or plans for pregnancy; breastfeeding.

Arms A1 and A3 are non-randomized and open label. Arm B is randomized and double blind.